Serre’s Conjectures

Henri Darmon

September 9, 2007

Contents

1 Statement of Serre’s conjecture 2

2 Serre’s recipe for N(ρ), k(ρ) and ε(ρ) 4

3 Evidence for Serre’s conjecture 73.1 Proofs of Serre’s epsilon-conjecture . . . . . . . . . . . . . . . 7

3.1.1 Theorems of Mazur and Ribet . . . . . . . . . . . . . . 73.1.2 The latest word . . . . . . . . . . . . . . . . . . . . . . 133.1.3 Raising the level . . . . . . . . . . . . . . . . . . . . . 14

3.2 Cases where the Serre conjecture is known . . . . . . . . . . . 153.2.1 Cases where ρ has dihedral image . . . . . . . . . . . . 153.2.2 Cases where F = F3 . . . . . . . . . . . . . . . . . . . 16

4 Applications 174.1 Diophantine applications: Fermat’s Last Theorem and some

variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Relation with the Shimura-Taniyama conjecture . . . . . . . . 22

5 Wiles’ work and the Serre conjectures 23

1

This article explains Serre’s conjectures relating mod p Galois represen-tations of Gal(Q/Q) to modular forms mod p, with special emphasis onthe aspects related to Wiles’ recent breakthrough on the Shimura-Taniyamaconjecture.

It is really impossible to improve on Serre’s original exposition, given in[Se7]. The reader is urged to consult [Se7] before reading this article.

It is a pleasure to thank Fred Diamond and Eric Liverance for many use-ful discussions over the last year related to the topics of this paper, andthe anonymous referee for making a careful and thorough review of themanuscript.

1 Statement of Serre’s conjecture

We begin with a statement of Serre’s conjectures. For generalities on modularforms, see [Sh] or the paper by Diamond and Im in this volume.

Letρ : GQ −→ GL2(F)

be an irreducible two-dimensional representation of GQ = Gal(Q/Q) over afinite field F of characteristic p. Asume that ρ is odd, i.e., det ρ : GQ −→ F∗

is an odd character. This means that if c is a complex conjugation, then ρ(c)has eigenvalues 1 and −1.

Note that, if ρ is unramified at l, and Frobl is a Frobenius element at l,then ρ(Frobl) is a well-defined conjugacy class in GL2(F); in particular, itscharacteristic polynomial is well defined.

If R is any subring of C, let Sk(N, ε, R) be the space of cusp forms ofweight k, level N , and character ε with Fourier coefficents in R. These arethe functions on the upper half plane which vanish at the cusps, satisfy thetransformation property

f(aτ + b

cτ + d) = (cτ + d)kε(d)f(τ),

for all

(a bc d

)∈ Γ0(N), and can be written in the form

f(τ) =∑n≥1

anqn, q = e2πiτ , an ∈ R.

2

If R contains the ring Z[ε] generated by the values of the character ε, thenwe have (q-expansion principle)

Sk(N, ε, R) = Sk(N, ε,Z[ε])⊗R.

For any ring R equipped with a map φ : Z[ε] −→ R, we can thus consistentlydefine

Sk(N, ε, R) := Sk(N, ε,Z[ε])⊗φ R.

The Hecke operators Tn with gcd(n, N) = 1 and Uq with q|N act on the spacesSk(N, ε, R) and on the subspace Snew

k (N, ε, R) of newforms. A simultaneouseigenform for this commuting algebra of operators will simply be called aneigenform, and will be said to be normalized if its first Fourier coefficient a1

is equal to 1.We say that ρ is modular if there exists a normalized eigenform f (of some

weight k ≥ 2, level N , and character ε) with Fourier coefficients in F,

f =∑n≥1

anqn, a1 = 1, an ∈ F,

such that for all l which are unramified for ρ and do not divide Np, ρ(Frobl)has characteristic polynomial

x2 − alx + lk−1ε(l).

In this case, we say that ρ and f are associated . A construction of Eichler andShimura for weight 2, and Deligne in weight k ≥ 2, shows that any eigenformf gives rise to an associated (not necessarily irreducible) representation ρ. In[Se7], Serre conjectures that the converse holds as well. In some sense, this isan analogue of the Shimura-Taniyama conjecture for mod p representations.

Conjecture 1.1 (Serre’s conjecture, vague form) Any odd irreduciblerepresentation ρ as above is modular.

Serre’s conjecture is much more precise than this; that is what accounts forits usefulness and importance. In fact, Serre gives a precise recipe (describedin sec. 2) for assigning to ρ a weight k(ρ) > 0, a level N(ρ) > 0, and anF-valued character ε(ρ). He then conjectures that

Conjecture 1.2 (Serre’s conjecture, precise form) There exists a nor-malized mod p eigenform of level N(ρ), weight k(ρ), and (when char F > 3)character ε(ρ) which is associated to ρ.

3

Remark: When char F = 2 or 3, it is not always possible to find an eigenformof the correct weight, level, and character associated to ρ. The difficulty isdue to the possible presence of elliptic points of order 2 or 3 on the modularcurves X0(N). The “naive” definition of modular forms mod p that we areusing (following Serre) is not quite adequate in this context, and Katz’sdefinition of modular forms mod p is more appropriate for dealing with thesesituations. (cf., for example, [Ed].)

The outline of this paper is as follows. Section 2 partly explains Serre’srecipe for N(ρ), k(ρ), and ε(ρ), completing the statement of conj. 1.2.

Section 3 presents some of the evidence for Serre’s conjecture. The evi-dence that exists is of three types. Firstly, computational evidence has beenamassed, mostly by Mestre, in support of conj. 1.2. Secondly, a great dealof work has been done in the direction of proving that conjecture 1.1 im-plies conj. 1.2. Thirdly, (and this is a key point in Wiles’ general attack onthe Shimura-Taniyama conjecture) the Serre conjecture is largely known tobe true when F is the field F2 with two elements, when the image of ρ isdihedral, or when F is the field F3 with three elements, the last thanks tothe work of Langlands and Tunnell on base change. Much of the computa-tional evidence is summarized in [Se7], §5, and we will not say more on this,focusing instead on the theoretical evidence.

Sec. 4 is devoted to various applications of the Serre conjectures to Fer-mat’s Last theorem and other Diophantine questions (see also the article byLiem Mai in this volume), and to the Shimura-Taniyama conjecture.

We conclude in section 5 by briefly mentioning the relation between Wiles’work and the Serre conjectures.

2 Serre’s recipe for N(ρ), k(ρ) and ε(ρ)

The invariant N = N(ρ) attached to ρ is the Artin conductor of the represen-tation ρ, with the possible factors of p removed; see [Se7], §1.2 for the precisedefinition. In particular, the level N(ρ) is divisible only by the primes l 6= pwhere ρ is ramified, and the value of N(ρ) depends only on the restrictionof ρ to the decomposition groups Dl at these ramified places. Note that, bydefinition, the level N(ρ) is always prime to p, although the representationρ may be ramified at p (and, in fact, typically is).

The character ε(ρ) is read off from the determinant character det ρ :

4

GQ −→ F∗ associated to ρ, as follows: a direct calculation shows that theconductor of det ρ divides Np, so that det ρ can be identified with an F-valued Dirichlet character (Z/NpZ)∗ −→ F∗, or, by the Chinese remaindertheorem, with a pair of characters:

ε : (Z/NZ)∗ −→ F∗, φ : (Z/pZ)∗ −→ F∗.

We set ε(ρ) := ε.

We can writeφ(x) = xk0−1, 2 ≤ k0 ≤ p.

The integer k0 determines the value of k(ρ) mod p− 1, namely, we have:

k(ρ) ≡ k0 (mod p− 1).

For the precise value of k(ρ), one needs to shift the value of k0 up by a certainmultiple of (p − 1). This corresponds to predicting the filtration (cf. [Se2]),and not just the weight, of the corresponding modular form mod p.

The recipe for the precise value of k(ρ) depends on the value of certainexponents a and b associated to the restriction of ρ to the inertia group Ip atp. For simplicity we confine ourselves to the case where p is odd, referringthe reader to [Se7] for the complete recipe. Let Wp denote the wild inertiasubgroup (which is the maximal pro-p-subgroup of Ip), and let V be thetwo-dimensional F-vector space which realizes ρ. The quotient It = Ip/Wp isisomorphic to lim←F∗pn , where the inverse limit is taken with respect to thenorm maps. It can be shown that Wp acts trivially on the semi-simplificationV ss of V , so that Ip acts on V ss via its tame quotient It. Since It is abelian,this action is reducible and corresponds to two characters φ and φ

′of It

with values in F∗p. The fact that the representation ρ extends to the full

decomposition group Dp shows that the characters φ and φ′are stable under

the action of Frobenius x 7→ xp. Hence we can distinguish two cases:Case 1: φp = φ

′, φ

′p= φ. Then we can write

φ = Ψa+pb = ΨaΨ′b,

where Ψ : It −→ F∗p2 is one of the two natural projections. We normalize theexponents a and b so that

0 ≤ a, b ≤ p− 1.

5

Case 2: φp = φ, φ′p = φ

′. Then we can write

ρ|Ip =

(χa ∗0 χb

),

where χ : It −→ F∗p is the natural map (i.e., the cyclotomic character). Theexponents a and b are well-defined modulo p− 1, and we normalize them sothat

0 ≤ a ≤ p− 2 if ρ|Ip is semisimple, 1 ≤ a ≤ p− 1 otherwise.

0 ≤ b ≤ p− 2.

Now the formula for k is

k = 1 + a + b + (p− 1) min(a, b) + (p− 1)δ,

where δ = 0 or 1, the case δ = 1 arising when (a, b) = (0, 0) (i.e., ρ isunramified at p) or when ρ|Ip is “tres ramifie”, c.f. [Se7], §2.4. For moredetails the reader is invited to consult §2 of [Se7] or §4 of [Ed]. Serre’s preciserecipe for the weight took shape through an exchange of letters with J-M.Fontaine; Fontaine’s ideas have been crucial in elucidating the relationshipbetween the weight and restriction to the inertia group at p of a modularmod p Galois representation.Remark: There is a certain amount of flexibility in defining N(ρ) and k(ρ).For example, if f is an eigenform of weight k and level N = Mpr, thenthe mod p representation ρ associated to f by Deligne’s construction alsoarises from an eigenform of level M and weight k

′for some k

′(cf. §2 of

[Ri4]). For an example where this occurs, see for instance [Se4], th. 11. The

representation ρ also arises from a form of weight 2 and level Mpr′for some

r′; cf. §6 of [Di2] and [Wi1].

An example: the Galois representations associated to a semi-stable ellipticcurve over Q: Let E be a semi-stable elliptic curve over Q and let ρE,p

ρE,p : GQ −→ Aut (Ep) ' GL2(Fp)

be the Galois representation associated to Ep. Let NE be the conductor ofE; since E is semi-stable, NE is simply the product of the primes of badreduction of E. Let ∆E be the minimal discriminant of E.

6

Proposition 2.1 The invariants N(ρE,p), k(ρE,p) and ε(ρE,p) are given by

1. N(ρE,p) is the product of all the primes l 6= p such that ordl(∆E) 6= 0(mod p).

2. k(ρE,p) = 2, if ordp(∆E) ≡ 0 (mod p), and is equal to p+1 otherwise.

3. ε(ρE,p) = 1.

Proof: See [Se7], prop. 5.Remark: In general, we say that ρ is finite at a prime l if, when l 6= p, ρ isunramified, and if, when l = p, ρ comes from a finite flat group scheme overZp. The condition ordp(∆E) ≡ 0 (mod p) implies that ρE,p is finite at p.

3 Evidence for Serre’s conjecture

3.1 Proofs of Serre’s epsilon-conjecture

Conjecture 1.2 appears very difficult to attack except in all but a few veryspecial cases. A more manageable problem has been to prove conj. 1.2,assuming that conj. 1.1 is satisfied, which is expressed in the following con-jecture (known as Serre’s “epsilon conjecture”).

Conjecture 3.1 (Serre’s epsilon-conjecture) If ρ is modular (i.e., is as-sociated to an eigenform mod p of some level, weight, and character), thenit is associated to a modular form mod p of level N(ρ), weight k(ρ), and (ifcharF > 3) character ε(ρ).

Alot has been proved in this direction, thanks to the work and ideas of manypeople, including N. Boston [BLR], H. Carayol [Ca], R. Coleman [CV] F.Diamond [Di1], [DT1], [DT2], [Di2], B. Edixhoven [Ed], G. Faltings, J-M.Fontaine, B. Gross [Gr], B. Jordan [JL], H.W. Lenstra, R. Livne, B. Mazur,K. Ribet [Ri2], [Ri4], J-P. Serre, R. Taylor, J. Tilouine, F. Voloch, and A.Wiles.

3.1.1 Theorems of Mazur and Ribet

The first result in the direction of Serre’s epsilon-conjecture was proved byB. Mazur:

7

Theorem 3.2 (Mazur) Suppose that ρ : GQ −→ GL2(F) is absolutely ir-reducible and arises from an eigenform of weight 2, level N , and trivial char-acter. If l||N but ρ is finite at l, and if

l 6≡ 1 (mod p),

then ρ arises from a mod p eigenform on X0(M), M = N/l.

The argument is reproduced in [Ri2].The next major breakthrough came with the work of Ribet, who showed

how to remove all primes l at which ρ is finite (and not just the l 6≡ 1(mod p)) from the level of the mod p representation.

Theorem 3.3 (Ribet) Suppose that ρ : GQ −→ GL2(F) is absolutely irre-ducible and arises from an eigenform of weight 2, level N , and trivial char-acter. Suppose also that char(F) is odd. If l||N but ρ is finite at l, then ρarises from a modular form of level M = N/l.

This result is enough (cf. 4.1, or the articles by Mai and Prasad in this vol-ume) to show that the Shimura-Taniyama conjecture implies Fermat’s Lasttheorem. Ribet proved his result by a very ingenious argument, exploiting adeep interplay between the arithmetic of modular curves and Shimura curvesassociated to indefinite quaternion algebras. For the details on the proof, seePrasad’s article in this volume, (or [Ri2], [Ri3], and [Ri4]).

The following examples give some illustrations of the theorems of Mazurand Ribet. We follow the notations of Cremona’s book [Cr], which extendsthe classical Antwerp tables [MF].

Examples: 1. Let E = X0(11) be the elliptic curve with equation

y2 + y = x3 − x2 − 10x− 20

having conductor N = 11 and discriminant ∆ = −115. Let ρ be the mod5 representation associated to E. Prop. 2.1 gives N(ρ) = 1, k(ρ) = 2, andε(ρ) = 1. By Ribet’s theorem (note that 11 ≡ 1 (mod 5)), if ρ were ir-reducible it would arise from an eigenform of weight 2, level 1, and trivialcharacter. Since there are no such forms, the representation ρ must be re-ducible. This, of course, is well known, and one does not require the fullpower of Ribet’s deep theorem to prove it! In fact, one knows that

E5 ' Z/5Z⊕ µ5

8

as a Galois module.

2. The curve 57C (or 57F in the Antwerp tables)

57C : y2 = x3 + x2 + 20x− 127/4 = f(x).

of conductor 57 = 3 · 19 has discriminant ∆ = −31019, and its mod 2 repre-sentation is irreducible. Hence the mod 2 representation ρE,2 arises from theunique cusp form of weight 2 on X0(19). There is a unique isogeny class ofcurves of conductor 19, represented by the curve 19C with equation

19C : y2 = x3 + x2 + x + 1/4 = g(x).

It is not hard to check that these two curves define the same mod 2 rep-resentation, by showing that the polynomials f(x) and g(x) have the samesplitting field. If α denotes the real root of f(x) and β the real root of g(x),then

α = 4β2 − 3β.

In this example, the conclusion of Ribet’s theorem 3.3 holds, even thoughthe hypothesis char(F) 6= 2 is not satisfied. It is likely that this assumptioncan be removed, especially when F = F2.

3. The curve33A : y2 + xy = x3 + x2 − 11x

has conductor N = 33 and discriminant ∆ = 36112. Its mod 3 representationρE,3 is irreducible, and finite at 3, hence by Mazur’s result it arises from a formof weight 2 and level 11. There is only one such eigenform, correspondingto the curve X0(11), and one checks that their Fourier coefficients al are thesame mod 3 when l 6= 3, 11, at least for l ≤ 43:

l 2 3 5 7 11 13 17 19 23 29 31 37 41 4333A 1 −1 −2 4 1 −2 −2 0 8 −6 −8 6 −2 011A −2 −1 1 −2 1 4 −2 0 −1 0 7 3 −8 −6

4. The curve46A : y2 + xy = x3 − x2 − 10x− 12

is a modular elliptic curve of conductor N = 2 · 23 and discriminant ∆ =−210 · 23. Since the mod 5 representation associated to E is irreducible, it

9

must be associated to a mod 5 eigenform of weight 2 on X0(23). There areno rational eigenforms of weight 2 on X0(23), but the theta functions

θ1 =∑

m,n∈Zqm2+mn+6n2

(= 1 + 2q + 2q4 + 4q6 + 4q8 + 2q9 + · · ·)

θ2 =∑

m,n∈Zq2m2+mn+3n2

(= 1 + 2q2 + 2q3 + 2q4 + 2q6 + 2q8 + 2q9 + · · ·)

associated to the two classes of binary quadratic forms of discriminant −23give modular forms of weight 1 on X0(23) with character ( ·

23) (cf. [Hc]).

(Note that

1

2(θ1 − θ2) = η(τ)η(23τ) = q

∏(1− qn)(1− q23n)

is a cusp form of weight 1 on X0(23) with character.) Setting

F =1

2(θ1 − θ2)θ1 = q + q2 − 3q3 − 2q4 + 2q5 − q6 + 4q7 − 3q8 + 2q9 + · · ·

G =1

2(θ1 − θ2)θ2 = q − q2 + q3 − 2q5 − 3q6 + q8 + 2q9 + 4q10 + · · ·

gives a Q-basis for the space of cusp forms of weight 2 on X0(23). The actionof the Hecke operator T2 can be computed explicitly and is given by:

T2F =1

2(F + G), T2G =

1

2(F − 3G).

By diagonalizing T2, (letting ω = 1+√

52

be the golden ratio, and ω its conju-gate) we find that the form

g =1

2(ωF + (1 + ω)G) = q − ωq2 −

√5q3 − ωq4 − 2ωq5 − (ω + 2)q6 + · · ·

is an eigenform of weight 2 for X0(23) with trivial character. (The othereigenform, of course, is merely the Galois conjugate.) One can check thatthe first few Fourier coefficients al(f) of f are congruent to the coefficientsal(g) of g modulo the ideal (

√5) when l 6= 2, 23:

l 2 3 5 7 11 13 17 19al(f) −1 0 4 −4 2 −2 −2 −2al(g) −ω −

√5 −1 +

√5 1 +

√5 −3−

√5 3 3−

√5 −2

10

5. The modular elliptic curve

988B : y2 = x3 − 362249x + 165197113

listed in Cremona’s tables has conductor N = 988 = 22 · 13 · 19 and discrim-inant

∆ = −24 · 13 · 1913.

The Galois representation ρ13 acting on the 13-division points of E is un-ramified at 19, and Mazur’s theorem says that ρ13 is associated to a modularform mod 13 of level 52. There is a unique rational eigenform of level 52,given by the curve

52A : y2 = x3 + x− 10,

and one finds that the Fourier coefficients al (l not dividing N) associated tothese two curves agree mod 13, at least for l ≤ 43:

l 2 3 5 7 11 13 17 19 23 29 31 37 41 43988B 0 0 2 −2 −2 −1 −7 1 −5 2 −3 7 7 −952A 0 0 2 −2 −2 −1 6 −6 8 2 10 −6 −6 4

This gives an example of two elliptic curves over Q whose mod 13 Galoisrepresentations are isomorphic. It would be interesting to see how often suchpairs occur. (See the discussion is sec. 4.1.)

6. This example examines what happens when one replaces mod p represen-tations by mod pn representations.

The curves

142A : y2 + xy + y = x3 − x2 − 12x + 15,

142E : y2 + xy = x3 − x2 − 2626x + 52244,

(denoted 142F and 142G respectively in the Antwerp tables) have discrim-inant 29 · 71 and 227 · 71 respectively. By using Tate’s analytic descriptionof these curves over Q2 (cf. Liem Mai’s article in this volume), one can seethat the mod 9 representation ρ1 associated to 142A, and the mod 27 rep-resentation ρ2 associated to 142E, are unramified at 2. A natural extensionof Ribet’s theorem, replacing mod p representations with mod pn represen-tations, would lead us to expect that ρ1 (resp. ρ2) is realized on the pointsof order 9 (resp. 27) of the Jacobian of the modular curve X0(71).

11

The genus of X0(71) is 6. One constructs modular forms of weight 2 andlevel 71 as in example 4, by letting

θ1 =∑

m,n∈Zqm2+mn+18n2

(= 1 + 2q + 2q4 + 2q9 + 2q16 + 4q18 + · · ·),

θ2 =∑

m,n∈Zq4m2+3mn+5n2

(= 1 + 2q4 + 2q5 + 2q6 + 2q12 + 2q15 + 2q16 + · · ·),

be the theta functions corresponding to the two classes of quadratic forms ofdiscriminant −71, and setting

F1 = θ1(θ1 − θ2)/2, Fi+1 = T2(Fi).

One checks from the q-expansions that F1, . . . , F6 are linearly independent,and hence generate the space of cusp forms of weight 2 on Γ0(71). Further-more,

T2(F6) = 9F1 − 3F2 − 23F3 + 5F4 + 9F5 − F6.

By diagonalizing the Hecke operators T2, one finds that the eigenforms forthe Hecke algebra are given by the forms:

(15−3α2)F1+(20−3α−4α2)F2+(α2−4α−8)F3+(α2+α−9)F4+(α+1)F5+F6,

(6−3α2)F1+(2α2−3α+2)F2+(5α2+5α−17)F3−(α2+3)F4−(α2+α−4)F5+F6,

where α is one of the three roots of the equation α3 − 5α + 3 = 0. Bynormalizing these forms so that the coefficient of q is 1, one obtains the qexpansions:

f = q + αq2 + (−α2 + 3)q3 + (α2 − 2)q4 + (−α− 1)q5 + (−2α + 3)q6 + · · · ,

g = q +(−α2−α+3)q2 +(α2 +α− 3)q3 +(α+1)q4 +(−α2− 2α+5)q5 + · · ·

The eigenforms f and g, together with their Galois conjugates, give the 6normalized newforms for X0(71).

The ideal (3, α) is the unique prime ideal of degree 1 in the ring Z[α]lying above 3. Let R ' Z3 be the completion of Z[α] at this prime ideal, andlet f and g be the images of the forms f and g in S2(71, R). Their Fouriercoefficients for the first few primes are listed modulo 81, (i.e., with a 3-adicaccuracy of 3−4) in the following table:

12

l 2 3 5 7 11 13 17 19 23 29f 60 48 20 24 57 4 24 73 68 70g 69 12 11 24 39 40 15 37 77 34

From the table it appears that al(f) ≡ al(g) (mod 9). Assuming that thisis true, one can see that the forms

fA =7f − 4g

3(mod 9), fE = 2f − g (mod 81)

are modular forms in S2(71,Z/9Z) and S2(71,Z/81Z) which are eigenformsfor the Hecke operators. From the following table, one checks that the Fouriercoefficients of the form corresponding to 142A are congruent to those of fA

mod 9, at least for l 6= 2, l ≤ 29:

l 2 3 5 7 11 13 17 19 23 29fA 3 6 5 6 0 1 0 4 2 1

142A 1 −3 −4 −3 0 1 0 −5 −7 −8

Likewise, one checks that the Fourier coefficients of the form correspondingto 142E are congruent to those of fE mod 27, at least for l 6= 2, l ≤ 29:

l 2 3 5 7 11 13 17 19 23 29fE 51 84 29 24 75 49 33 28 59 25

142E −1 3 2 −3 −6 −5 6 1 5 −2

This example suggests that the philosophy of Serre’s conjectures, and of theε-conjecture, extends to mod pn representations. Wiles has proved a numberof precise statements in this direction, and used them to bound the order ofthe Selmer group of the symmetric square under certain conditions; cf. [Wi1],or [Wi2].

3.1.2 The latest word

The work of Mazur and Ribet alluded to before was mainly concerned withmodular forms of weight 2, and trivial character. A great number of math-ematicians have extended the scope of these results, to cover more generalcases involving arbitrary weights, levels, and characters, so that now the full

13

epsilon conjecture is almost proved. For a good summary of these results,together with an explanation of the techniques involved in proving them, andan extensive bibliography, see [Ri4] and [Di2].

The latest result, which is the culmination of all these efforts, is provedin [Di2]: say that the irreducible representation ρ is an exceptional case ifcharF = 3 and ρ is induced from a character of Gal(Q/Q(

√−3)), or if

charF = 2 and ρ is induced from a character of Gal(Q/Q(i)).

Theorem 3.4 Assume F is a field of odd characteristic. If ρ : GQ −→GL2(F) is a representation arising from an eigenform, then ρ is associatedto an eigenform of level N(ρ), weight k(ρ), and, if ρ is not an exceptionalcase, character ε(ρ).

For more details, see [Di2].

3.1.3 Raising the level

The results alluded to so far in this section have to do with “lowering thelevel” of a modular Galois representation; i.e., showing that if it arises from amodular form of some level it also arises from the “optimal” level N(ρ) pre-dicted by the Serre conjectures. There is a considerable amount of literatureon congruences between modular forms which is devoted to the problem oflisting the possible levels of newforms which are congruent mod p to a giveneigenform f . For example, one has the following result which is a corollaryof [Ri1] and [Ca]:

Theorem 3.5 Let f be a newform of weight 2, trivial character, and levelN , and assume that the associated representation ρ is irreducible and notan exceptional case. Suppose that l is a prime not dividing N and that (l −1)(al(f)2−(l+1)2) is divisible by a prime P over p. Then there is a newformg of weight 2, trivial character and level dl for some d|N such that g iscongruent to f mod P.

For more precise results in this direction, see [DT1].There are also more precise quantitative measures of the “amount” of

mod p congruences that arise between f and newforms of level Nl, involvingthe notion of the “congruence ideal” of a Hecke ring. For precise definitions,see Kumar Murty’s article in this volume. This shows that (in some “sophis-ticated” sense) the “number” of eigenforms of some level M divisible by N

14

which are congruent to f mod p can be described by a simple formula. Thisremark plays a key role in Wiles’ proof of the Shimura-Taniyama conjecturefor infinitely many j-invariants.

3.2 Cases where the Serre conjecture is known

In spite of the spectacular success in establishing more and more cases ofSerre’s epsilon-conjecture, very little is known about conj. 1.2 without firstassuming conj. 1.1. There are a few notable exceptions, which play an im-portant role in Wiles’ work on the Shimura-Taniyama conjecture.

3.2.1 Cases where ρ has dihedral image

Suppose that the image of ρ is isomorphic to a dihedral group D2n with(n, p) = 1.

The group D2n can be embedded in GL2(C), and hence ρ gives rise toan Artin representation

ρ′: GQ −→ GL2(C).

Since ρ is odd, we may assume that ρ′is odd. This representation is asso-

ciated to a cusp form g of weight 1, which can be constructed explicitly interms of theta-functions associated to (definite or indefinite) binary quadraticforms, as was known already to Hecke.

The construction works as follows. The field L cut out by ρ is an abelianextension, with Galois group a cyclic group G of order n, of a quadratic fieldK/Q, and we can write

ρ′= IndK/Qχ,

where χ : Gal(L/K) −→ C∗ is a non-trivial one-dimensional character, whichcan be viewed as a character on the ideals of OK by class field theory.

Let τ be a reflection in Gal(L/Q), and let D = Disc(Lτ ). Let

θ =∑J

χ(J)qNJ

be the theta function, which is a cusp form of weight 1 and level |D|.By multiplying θ by an Eisenstein series of weight 1, level p and character

ω−1, where ω is the Teichmuller character, one obtains a form f of weight 2which is an eigenform for the Hecke operators Tl mod p. Hence we have:

15

Proposition 3.6 If ρ has dihedral image, then ρ is associated to a modularform, i.e., conj. 1.1 is true for ρ.

Furthermore, thm. 3.4 tells us that:

Corollary 3.7 If ρ is dihedral and charF is odd, then ρ is associated to amodular form of level N(ρ), weight k(ρ), and, if ρ is not an exceptional case,character ε(ρ).

An interesting special case is the one where F = F2. Here the imageis contained in S3 which is a dihedral group, so conj. 1.1 is known for ρ,but Serre’s epsilon-conjecture remains unproved. The status of conj. 1.2 forF = F2 is therefore unclear at present, although it may be quite accessible,since in some cases the epsilon conjecture in the dihedral case can be provedwithout appealing to thm. 3.3.

3.2.2 Cases where F = F3

When F = F3, the group GL2(F3) can be embedded into GL2(C), allow-ing one to lift the mod 3 representation to a characteristic 0 representationρ

′. Moreover, the image of ρ

′is a solvable group: the group GL2(F3) is

isomorphic to a double cover of the alternating group A4. A deep result ofLanglands and Tunnell [La], [Tu] shows that ρ

′is associated to a modular

form of weight 1. By the same trick of multiplying this form by an appro-priate Eisenstein series, one can exhibit a modular form of higher weightassociated to ρ, and show that ρ satisfies conj. 1.1. In light of thm. 3.4, wetherefore have:

Theorem 3.8 If ρ : GQ −→ GL2(F3) is absolutely irreducible, then it isassociated to an eigenform of level N(ρ), weight k(ρ), and, if ρ is not in theexceptional case, character ε(ρ).

This theorem is at the center of Wiles’ very compelling strategy for provingthe Shimura-Taniyama conjecture for semi-stable elliptic curves. See forexample Kumar Murty’s article in this volume.

16

4 Applications

4.1 Diophantine applications: Fermat’s Last Theoremand some variants

Letap + bp = cp, abc 6= 0, p ≥ 5

be a solution to Fermat’s equation. Assume without loss of generality thata ≡ −1 (mod 4) and that b is even, and let

E : y2 = x(x− ap)(x + bp)

be the elliptic curve first considered by Hellegouarch [He]. It can be shownthat E is a semi-stable elliptic curve and that its discriminant ∆E is

∆E = −2−8(abc)2p.

(cf. [Se7], p. 200). Consider the mod p representation

ρ : GQ −→ Aut (Ep) ' GL2(Fp)

associated to Ep. By a theorem of Mazur [Ma1], ρ is irreducible. Prop.2.1 implies that N(ρ) = 2, k(ρ) = 2, and ε(ρ) = 1, contradicting conj.1.2, since there are no non-trivial eigenforms of weight 2 on Γ0(2). Thus,Serre’s conjecture implies Fermat’s Last Theorem. (For more details, seeMai’s article in this volume, or [Se7], §4.)

In fact, thanks to Ribet’s work on the epsilon-conjecture 3.1, conj. 1.1applied to the mod p Galois representation ρ is already enough to implyFermat’s Last Theorem. If E is a modular elliptic curve, then ρ satisfiesconj. 1.1. Hence Fermat’s Last Theorem follows from the Shimura-Taniyamaconjecture (for semi-stable elliptic curves), confirming a remarkable insightof G. Frey.

The reader should consult §4 of [Se7] for more examples where the Serreconjectures are used to study certain variants of the Fermat equation, forexample, equations of the form

Axp + Byp = Czp.

17

It is interesting to explore the limits of the Serre conjectures in studyingDiophantine equations of the above type. Having come tantalizingly closeto Fermat’s Last Theorem, a good testing ground for further applications ofSerre’s conjecture (as well as a nice source of concrete Diophantine questions)is given by the following “generalized Fermat conjecture” [DG].

Conjecture 4.1 The equation

xp + yq = zr,1

p+

1

q+

1

r< 1, gcd(x, y, z) = 1, xyz 6= 0,

has no integer solutions except for

1 + 23 = 32, 25 + 72 = 34, 73 + 132 = 29, 27 + 173 = 712, 35 + 114 = 1222,

177+762713 = 210639282, 14143+22134592 = 657, 92623+153122832 = 1137,

438 + 962223 = 300429072, 338 + 15490342 = 156133.

So far, very little is known about this conjecture, which combines the diffi-culties of the Fermat and Catalan conjecture. However, one does have thefollowing fragments:

Proposition 4.2 If Serre’s conjecture 1.2 holds, then the equations

xp + yp = z2, p ≥ 13, p ≡ 1 (mod 4),

xp + yp = z3, p ≥ 13, p ≡ 1 (mod 3),

x4 − y4 = zp, p ≥ 13, p ≡ 1 (mod 4),

have no solutions (x, y, z) with xyz 6= 0 and gcd(x, y, z) = 1.

To prove these propositions one constructs the appropriate Frey curves as-sociated to solutions to the above equations, and uses Serre’s conjecture toprove that the associated mod p representation does not exist. Since theserepresentations come from elliptic curves, the Shimura-Taniyama conjectureis enough to deduce the result, in light of Ribet’s thm. 3.3. For the details,see [Da1] and [Da2].

A more interesting example is that of the equation

x4 + y4 = zp, gcd(x, y, z) = 1,

18

which generalizes the equation x4 + y4 = z2 originally considered by Fermat.Given a solution a4 +b4 = cp to this equation, one considers the elliptic curveover Q(i)

E : y2 = x3 + 4(1 + i)bx2 + 4i(b2 + ia2)x

which has discriminant∆ = 212cp(a2 − ib2).

The map η given by

η(x, y) =

(iy2

2x2,−4y(b2 + ia2)− iyx2

(2i− 2)x2

)

is a 2-isogeny from E to its Galois conjugate E′which is defined over Q(i).

The curve E is a Q-curve, i.e., it is an elliptic curve defined over a numberfield which is isogenous to all of its Galois conjugates. Even though E is notdefined over Q, it can be used to construct a 2-dimensional representationof Gal(Q/Q) as follows: Let η

′: E

′ −→ E be the Galois conjugate of ηover Q(i). One checks that ηη

′and η

′η are the endomorphisms of E

′and E

respectively given by multiplication by 2. If we set V = Ep × E′p, for p an

odd prime, then V is equipped with a natural action of Fp[φ] ' Z[√

2]⊗Fp,where φ : V −→ V is the endomorphism defined by

φ(P, Q) = (η′Q, ηP ).

In this way V is a module over Fp[φ] of rank 2. The natural action of GQ(i)

on V can be extended to an action of GQ, by defining

σ(P, Q) = (P σ, Qσ) if σi = i,

σ(P, Q) = (Qσ, P σ) if σi = −i.

This GQ-action commutes with the scalars in Fp[φ], and hence gives rise toa two-dimensional representation

ρ : GQ −→ Aut Fp[φ](V ) ' GL2(Z[√

2]⊗ Fp).

Because ordQ(∆) ≡ 0 (mod p) for all primes Q of Z[i] which are not above2, and because E is semistable at those primes, one sees that the represen-tation ρ is unramified for all primes 6= 2, p, and that it is finite at p. A morecareful analysis shows that

N(ρ)|214, k(ρ) = 2, ε(ρ) = 1.

19

One can hope to use the Serre conjecture to show that a mod p representationwith such a small conductor does not exist; this would give a Diophantineapplication of the Serre conjectures which does not pass through the Shimura-Taniyama conjecture.

Note that this application does follow from a somewhat stronger versionof the Shimura-Taniyama conjecture, which characterizes the elliptic curvesover Q which are modular. See the next section for details.

Although the Serre conjectures have striking Diophantine consequencesfor the Fermat equation and some variants, it seems that they do not yieldsweeping Diophantine results applying, say, to all of the equations of the typeAxp + Byp = Czp. For example (cf. [Se7], p. 204) the Serre conjectures donot allow one to show (or at least, not in an obvious way!) that the equation

xn + yn = 31zn, gcd(x, y, z) = 1, n ≥ 4.

has finitely many solutions (x, y, z, n). (Although this is certainly expectedto be true, and follows, for example, from the abc-conjecture.) The problemhere is that there is at least one non-trivial solution, (−1, 2, 1, 5), so that themethods based on Serre’s conjecture, which tend to prove non-existence ofsuch solutions, are bound to fail here.

To reap further Diophantine results from Serre’s conjecture, one needsmore knowledge about the Galois representations arising from elliptic curves.For example, Frey [Fr3] has made the following conjecture:

Conjecture 4.3 (Frey) Let A be an elliptic curve over a number field K.There are only finitely many pairs (E, p) consisting of an elliptic curve Eover K which is not isogenous to A and a prime number p > 5, such that

Ep ' Ap as GK −modules.

Remarks:1. For fixed p > 5, the conjecture is true, by Falting’s proof of the the Mordellconjecture. This is because pairs (E, p) as above correspond to rational pointson a twist of the modular curve X(p), which has genus greater than 1.2. The obvious analogue of the above conjecture with (E, p) replaced by pairs(f, p) where f is a modular form of weight 2 is of course false, as follows fromthm. 3.5.

20

One can propose even more ambitious conjectures. Say that an integer nhas the isogeny property (relative to a number field K) if the implication

An ' Bn as GK −modules ⇒ A is isogenous to B (1)

holds for all pairs of elliptic curves A, B over K. It is not known whetherthere are any integers satisfying the isogeny property (over Q, say), andexample 5 of sec. 3.1.1 shows that 13 does not have the isogeny property. Itis tempting, however, to conjecture the following:

Conjecture 4.4 Given any global field K, there exists a constant MK suchthat all n ≥ MK have the isogeny property.

This conjecture, which can be viewed as a “mod p” analogue of Tate’s isogenyconjecture proved by Faltings, seems very difficult to prove.

Remark: Say that n satisfies the weak isogeny property if the implication (1)holds, with at most finitely many exceptional pairs (A, B). A strengtheningof conj. 4.4 is

Conjecture 4.5 There exists an absolute constant M such that all n ≥ Mhave the weak isogeny property over all number fields K.

It would be very interesting to formulate a convincing guess about the precisevalue of M .

As Frey has observed, we have:

Proposition 4.6 If Serre’s conjecture 1.2 and Frey’s conjecture 4.3 hold,then the equation

Axn + Byn = Czn, n > 3, gcd(x, y, z) = 1,

has only finitely many integer solutions (x, y, z, n).

Sketch of proof: We argue by contradiction. Suppose that there are infinitelymany solutions (xi, yi, zi, ni). We can assume without loss of generality thatthe ni are distinct primes which do not divide 2ABC. Now, let

Ei : Y 2 = X(X − Axnii )(X + Bxni

i )

21

be the Frey curve associated to the solution (xi, yi, zi, ni), and let ρi be itsassociated mod ni representation. The level of ρi can be shown to divide32(ABC)2. By the Serre conjecture, each ρi arises from a mod ni eigenformfi of level dividing 32(ABC)2. Since there are finitely many such eigenforms,there is an eigenform f such that al(f) ≡ al(fi) (mod Ni) for infinitelymany i, where Ni is a place of Q above ni. We claim that f has integerFourier coefficients, contradicting Frey’s conjecture 4.3. For, let l be a primenot dividing 2ABC, and let p(x) be the minimal polynomial of al(f). Thecurve Ei has either semistable or good reduction at l. In the former case,al(f) ≡ ±(l + 1) (mod Ni), and in the latter, al(f) ≡ al(fi) (mod Ni). Bythe Hasse bound, al(fi) is an integer of absolute value less than 2

√l. Hence

there exists a in the finite set

{0,±1,±2, . . . ,±[2√

l],±(l + 1)}

such thatal(f) ≡ a (mod Ni) for infinitely many i.

Hence p(a) ≡ 0 (mod ni) for infinitely many i, so that p(a) = 0. This showsthat al(f) = a is rational, and concludes the proof.

4.2 Relation with the Shimura-Taniyama conjecture

Letf(τ) =

∑n≥1

anqn, q = e2πiτ , a1 = 1

be a normalized eigenform of weight 2 on Γ0(N) with trivial nebentypuscharacter. The differential f(τ)dτ is invariant under the action of Γ0(N); ifthe Fourier coefficients an belong to Z, then the function

φf (τ) := 2πi∫ τ

i∞f(z)dz

=∑n≥1

an

nqn

defines a complex-analytic map from the upper half plane H ∪ {cusps} toC/Λf , where Λf is a rank 2 lattice, generated by the modular symbols φf (

aNb

),for a, b ∈ Q with (a, Nb) = 1. The elliptic curve E = C/Λf can in fact bedefined over Q. It has good reduction at p for all p 6 |N , and

#E(Fp) = p + 1− ap.

22

The conjecture of Shimura Taniyama states that this beautiful connectionbetween eigenforms with rational Fourier coefficients and elliptic curves overQ also goes in the other direction, namely:

Conjecture 4.7 (Shimura-Taniyama) If E is an elliptic curve over Qof conductor N , then there exists a normalized eigenform f such that E isisogenous to C/Λf .

By the work of Eichler and Shimura, if E satisfies the Shimura-Taniyamaconjecture, then ρE,p satisfies conj. 1.1; we have already used this fact (moreor less implicitely) several times so far. In fact, it is also true that conj. 1.2implies the Shimura-Taniyama conjecture.

Theorem 4.8 Serre’s conj. 1.2 implies conj. 4.7.

The proof is explained in [Se7], §4.6, but the reader may find it instructiveto work it out on her own.

The Serre conjectures also imply generalizations of the Shimura-Taniyamaconjecture which say that every abelian variety with real multiplications is aquotient of J0(N) for some N (cf. [Se7], §4.7).

The following is also worth mentioning: suppose E is an elliptic curve overQ, which is a Q-curve. (cf. 4.1.) It is not hard to see that if E is modular(i.e., is a quotient of J0(N) for some N) then E is a Q-curve. Conversely:

Proposition 4.9 (Ribet) If Serre’s conjecture 1.2 is true, then an ellipticcurve E over Q is modular if and only if it is a Q-curve.

The proof is explained in [Ri5].

5 Wiles’ work and the Serre conjectures

We finish with some brief comments about the relation between Wiles’ workand the Serre conjectures, following [Wi1].

The Shimura-Taniyama conjecture states that the map

{Newforms of weight 2 on X0(N)with rational Fourier coefficients.

}−→

Isogeny classes ofelliptic curvesover Q of conductor N

23

is a bijection. One might try to tackle such a conjecture by showing that thetwo sets above have the same number of elements. One difficulty is that therationality condition on the Fourier coefficients is a very subtle one over whichone has little control. On the other hand, it is easy to count the number ofall eigenforms of weight 2 on X0(N). Such eigenforms (with not necessarilyrational coefficients) do not correspond to elliptic curves in general, but theydo give rise to p-adic Galois representations which generalize the Tate mod-ules Tp(E) of an elliptic curve, by the work of Eichler and Shimura. Assumefor simplicity that p2 does not divide N . It can be shown that the p-adicrepresentation ρ arising from an eigenform f of weight 2 and level N has thefollowing properties:

1. ρ is unramified outside Np.

2. (Weight condition) If l 6 |Np, the eigenvalues αl and αl of the Frobeniuselement ρ(Frobl) are the roots of a polynomial with coefficients in Z ⊂Zp, and, when viewed as complex numbers, they have absolute value√

l.

3. (Determinant condition) We have det(ρ) = χ, where χ : GQ −→ Z∗p isthe cyclotomic character.

4. (Condition at p) The restriction of the representation ρ to the inertiagroup Ip at p satisfies the condition of being ordinary or flat in Wiles’terminology. (For the definition of these terms, see Kumar Murty’sarticle in this volume.)

5. (Condition at∞) The representation ρ is odd, i.e., if c denotes complexconjugation, then ρ(c) has eigenvalues 1 and −1.

We call irreducible representations satisfying properties 1–5 admissible (in anon-standard terminology). One can define the conductor of ρ as Npδ, whereN is the Artin conductor of ρ and δ = 0 if ρ comes from a p-divisible groupover Zp, and δ = 1 otherwise.

One can generalize the Shimura-Taniyama conjecture to state that themap{

Newforms of weight 2on X0(N)

}−→

{Admissible p-adic representationsof GQ of conductor N

}

24

is a bijection. This conjecture is due, essentially, to Mazur.

Remark: By the isogeny conjecture of Tate which was recently proved byFaltings the functor E 7→ Tp(E) is a fully faithful functor from the categoryof isogeny classes of elliptic curves over Q to the category of admissible p-adicrepresentations of GQ. Hence, one does not lose any information in passingfrom the category of elliptic curves to the “larger” category of p-adic Galoisrepresentations.

Now the problem of proving the generalized Shimura Taniyama conjectureabove can be broken into two parts:

Problem 1: (Serre’s conjecture) Show that the map

{mod p newforms of weight 2on X0(N)

}−→

Odd, irreduciblemod p representationsof GQ of conductor N

is a bijection. Here the conductor of a mod p representation ρ is defined usingSerre’s recipe for N(ρ), explained in §2, except when k(ρ) > 2, in which caseone multiplies N(ρ) by an appropriate power of p.

Problem 2: (Lifting conjecture) Assuming that problem 1 is solved for aspecific mod p representation ρ0, i.e., there is an eigenform f0 such that f0

(mod p) is associated to ρ0. Show that the map:

Newforms f of weight 2on X0(N) such thatf ≡ f0 (mod p)

−→

Admissiblep-adic representationsρ of GQ

of conductor N such thatρ ≡ ρ0 (mod p)

is a bijection.

Wiles has made substantial inroads into problem 2, showing that thelifting conjecture is satisfied when p is odd and, for example:

1. The image of ρ0 is dihedral, or

2. ρ0 arises from a semistable, modular elliptic curve A and p does notdivide the degree of the modular parametrization X0(N) −→ A.

25

When p = 3 or 5, results of this kind are particularly interesting because thenfor a specific ρ0 there are infinitely many non-isomorphic elliptic curves Esatisfying ρE,p ' ρ0, and so this gives infinitely many distinct Q-isomorphismclasses of elliptic curves which are modular.

Remarks:1. The counting argument which was alluded to in the remark after thm.3.5 plays an important role in proving the second statement, by allowingWiles to reduce to the case where N = N(ρ0) in the statement of the liftingconjecture.2. The case p = 2 of Wiles’ program, which is not covered by the aboveresults, is also quite interesting: the desired upper bound on the Selmergroup of the symmetric square for p = 3, which is the major unresolvedissue in Wiles’ strategy, is known for the dihedral case, thanks to the workof Karl Rubin on the two variable main conjecture for quadratic imaginaryfields. Can one show that every (semistable, say) elliptic curve with a pointof order 2 over a non-cyclic cubic extension is modular? There seem to beno conceptual barriers in doing this, only technical difficulties (which couldstill make the task quite arduous!)

Wiles’ compelling strategy for proving the Shimura Taniyama conjecturebrilliantly avoids proving any new cases of the Serre conjecture. Rather, ituses the few cases where conj. 1.2 is known as a very tenuous foothold (“uneprise d’ongles”, in the words of Serre) from which to mount an impressiveattack on the Shimura Taniyama conjecture.

This means that the Serre conjectures remain wide open. These fascinat-ing conjectures, which represent a first step in the direction of a “Langlandsphilosophy mod p”, will probably keep number theorists busy in years tocome – perhaps long after the Shimura-Taniyama conjecture has been com-pletely proved.

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